All parabolas through three non-collinear points

Extract

If no two of three non-collinear points share the same x-coordinate, then the parabola y = a₂x² + a₁x + a₀ through the points is easily found by solving a system of linear equations. That is but one of an infinite number of parabolas through the three points. How does one find the other parabolas? In this note, we find all parabolas through any three non-collinear points by reducing the problem to finding the equation of a parabola by using rotations.

The parabola y = a₂x² + a₁x + a₀ has an axis of symmetry parallel to the y-axis. Other parabolas have an axis of symmetry that is parallel to some line y = mx. We focus on the angle θ that the axis of symmetry makes with the y-axis, as in Figure 1, so that tanθ = 1/m. To find the parabola associated with θ that goes through three non-collinear points, we rotate the three points counterclockwise by θ, find the equation of the parabola, and then rotate the parabola (and the three points) counterclockwise back by −θ so that the parabola goes through the original points.